A locking-free virtual element method for 3D linear elasticity problems

Abstract

This paper focuses on proposing and analyzing a new locking-free lowest order virtual element method for the linear elasticity problem in three dimensions. A virtual element function on a polyhedron $K$ is harmonic, while it is continuous piecewise linear corresponding to an auxiliary triangulation on the boundary $\partial K$. Such construction requires no further three-dimensional partition of $K$. Under some reasonable mesh assumptions, we derive the inverse inequality, the norm equivalence and the error estimate of the interpolation operator for the underlying virtual element. Using these results combined with a rigorous analysis, we establish a robust error estimate in $H^1$ norm for the proposed method. Finally, we perform numerical results to demonstrate theoretical findings.